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Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

M.I. Malkin

M.I. Malkin

  • Lobachevsky State University of Nizhniy NovgorodRussia
  • National Research University Higher School of EconomicsRussia
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K.A. Safonov

K.A. Safonov

  • Lobachevsky State University of Nizhniy NovgorodRussia
  • National Research University Higher School of EconomicsRussia
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Safonov, K.A.
  
16 lis 2020
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Data publikacji: 16 lis 2020
Zakres stron: 293 - 306
Otrzymano: 08 sty 2020
Przyjęty: 11 kwi 2020
DOI: https://doi.org/10.2478/amns.2020.2.00052
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© 2020 M.I. Malkin et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Afraimovich-Bykov-Shilnikov geometric model of Lorenz attractor
Afraimovich-Bykov-Shilnikov geometric model of Lorenz attractor

Fig. 2

Bifurcation diagram for the Shimizu-Morioka system near the point Q(α = 0.608, λ = 1.044) which corresponds to the homoclinic figure-8 bifurcation to a saddle with zero saddle value. (The green region corresponds to parameter values where the Lorenz attractor exists.)
Bifurcation diagram for the Shimizu-Morioka system near the point Q(α = 0.608, λ = 1.044) which corresponds to the homoclinic figure-8 bifurcation to a saddle with zero saddle value. (The green region corresponds to parameter values where the Lorenz attractor exists.)

Fig. 3

The graph of topological entropy for the logistic map fa(x) = ax(1 − x).
The graph of topological entropy for the logistic map fa(x) = ax(1 − x).

Fig. 4

The bifurcation diagram for map Tc,ɛ. Green region correspond to the existence of Lorenz attractor.
The bifurcation diagram for map Tc,ɛ. Green region correspond to the existence of Lorenz attractor.

Fig. 5

The kneading chart for map Tc,ɛ shows that the topological entropy as the function of c has a single minimum for ɛ in the interval [0,0.6] (more precise calculation gives ɛ ∈ [0,0.76]). Above the red line one has that Tc,ɛ is expanding (DTc,ɛ > 1), and there the topological entropy is monotone increasing in c.
The kneading chart for map Tc,ɛ shows that the topological entropy as the function of c has a single minimum for ɛ in the interval [0,0.6] (more precise calculation gives ɛ ∈ [0,0.76]). Above the red line one has that Tc,ɛ is expanding (DTc,ɛ > 1), and there the topological entropy is monotone increasing in c.

Fig. 6

The graph of Tc,ɛ and the trajectory of the discontinuity point for c > 2.
The graph of Tc,ɛ and the trajectory of the discontinuity point for c > 2.

Fig. 7

The second iteration of the map Tc,ɛ for ɛ = 0.7 with c = 1 (fig. a) and c = 1.1 (fig.b). In fig. b, a magnified fragment near the discontinuity point is shown.
The second iteration of the map Tc,ɛ for ɛ = 0.7 with c = 1 (fig. a) and c = 1.1 (fig.b). In fig. b, a magnified fragment near the discontinuity point is shown.

Fig. 8

Saddle-node bifurcation for ɛ = 0.7
Saddle-node bifurcation for ɛ = 0.7

Fig. 9

Trajectory of the discontinuity point at the boundary of the LA2 region for ɛ = 0.4.
Trajectory of the discontinuity point at the boundary of the LA2 region for ɛ = 0.4.

Fig. 10

Sketch of creation of the trivial lacuna
Sketch of creation of the trivial lacuna

Fig. 11

The graph of topological entropy with respect to parameter c
The graph of topological entropy with respect to parameter c

Fig. 12

The graph of topological entropy with respect to parameter ɛ
The graph of topological entropy with respect to parameter ɛ

Fig. 13

Illustration to theorem 4: above the dotted line one has the expanding condition, and in the whole green region there are no stable periodic orbits. PF curve indicates the pitchfork bifurcation.
Illustration to theorem 4: above the dotted line one has the expanding condition, and in the whole green region there are no stable periodic orbits. PF curve indicates the pitchfork bifurcation.
Język:
Angielski
Częstotliwość wydawania:
1 razy w roku
Dziedziny czasopisma:
Nauki biologiczne, Nauki biologiczne, inne, Matematyka, Matematyka stosowana, Matematyka ogólna, Fizyka, Fizyka, inne
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